Just feeding back on my own post: overall, I cannot say it was a negative experience having Mazda First handle my claim. They seem to be a well set up organization, with well-trained staff and clear processes. Definitely different from the numerous ambulance-chasers that called me from all over the country; how my contact details found their way to so many of these claims companies was quite a learning.
My claim was recorded over the phone at first, then through forms that I had to complete; most processing was online/electronic so minimal hassle from that perspective. Finding a repair company was slower than I thought, maybe because there aren't Mazda repair centres around where I live. Providing a replacement car for the duration also took a while, but the car was more than comparable so no complaints there.
The body shop itself was top-notch in terms of customer service, so I found it easier to liaise with them regarding progress and completion dates. As with many service organizations, Mazda First were slow on this front and I usually had better/more current information than they did. I guess the take-away is to stay on top of things and not trust them to be prompt; a cynic might say that the longer the claim processing lasted, the more they earn from it. Again, no complaints as such.
I was impressed that they had a Quality Assurance team, which handled an issue I had with the work. Basically, a third-party company of technical assessors approve/reject the work that the garage proposes, and this company initially rejected a piece of work because they thought it did not arise from the accident. It was all sorted out after a few calls and the car went back in for the remaining work, but again, caveat emptor - if I hadn't pushed it, it wouldn't have got done.
Bottom line - no regrets going through Mazda First. They are a bona fide organization with sensible processes and procedures.